BY MARTIN GARDINER, C.E. 115 



of an inscribable closed 3'gon whose sides pass in any order 

 through the three points o v o 9 , o 3 ." 



Now, if we assume the pole of the plane o o o g as a fourth 

 point, then evidently (see theorem 7) we have 



THEOREM 20. 



If there be a surface of the second degree and four points, such that 

 each one is the pole of the plane containing the of her three, then will 

 any point in the surface be an answerable position for the first extremity 

 of an inscribable closed 4<'gon whose sides pass in order through the 

 four points taken in any order. fjj^T The centre of the surface 

 and the three points at infinity indicated by the productions of 

 any system of conjugate diameters, are evidently four points such 

 that each one is the pole of the plane containing the other three. 



13. And from theorems 20 and 10 we have 



THEOREM 21. 



If there be a surface of the second degree and three points such 

 that the polar plane of each one contains the other two points, then will 

 the problem of the inscription of the closed 3'gons, the sides of which 

 pass in any order through these points, be partially porismatic. 

 fJSp These closed 3'gons will be imaginary when the trace of 

 the plane through the three points is imaginary. 



THEOREM 22. 



If there be a surface of the second degree and three points such 

 that the polar plane of each one contains the other two points, then 

 any point in the surface 'is an answerable position for the first ex- 

 tremity of an inscribable closed 6'gon whose first three sides and 

 whose second three sides pass through the three points taken in any 

 order. 



14. Now if we have a surface of the second degree and any 

 odd number n of points in one straight line, it is evident that the 

 points in which the straight line pierces the surface are answerable 



